On estimation of the PDF and the CDF of the one-parameter polynomial exponential family of distributions

In this article, we have considered the estimation of the probability density function and cumulative distribution function of the one-parameter polynomial exponential family of distributions. A number of probability distributions like the exponential, Lindley, length-biased Lindley and Sujatha are particular cases. Two estimators-maximum likelihood and uniformly minimum variance unbiased estimators of the probability density function and cumulative distribution function of the family have been discussed. The estimation issues of the length-biased Lindley and Sujatha distribution have been considered in detail. The estimators have been compared in mean squared error sense. Monte Carlo simulations and real data analysis are performed to compare the performances of the proposed estimators.